#1
What are the reasons for getting the tailpiece bigsby vs getting the one that has no tailpiece?
#3
a bigsby vibrato unit is a tailpiece. like WtrPlyr said, different bigsbys are designed to fit different guitars.
Rig Winter 2017:

Fender Jazzmaster/Yamaha SG1000
Boss TU-3, DS-2, CS-3, EHX small stone, Danelectro delay
Laney VC30-112 with G12H30 speaker, or Session Rockette 30 for smaller gigs
Elixir Nanoweb 11-49 strings, Dunlop Jazz III XL picks
Shure SM57 mic in front of the amp
#4
The top one is for guitar bodies that have curved tops, like most Les Pauls, the other one is for flat guitars, like most Gibson SG's.
Guitars:
Gibson SG Standard with Bigsby
Gibson Custom ES-137
Gibson Custom 54' Les Paul VOS Goldtop
Gretsch G6129T Silver Jet
1968 Harmony Rocket H75


Amps:
Fender Hot Rod Deluxe (modded)
1970 Fender Twin Reverb (blackfaced)